Finite Group Databasebeta

← C2×#22550093 #30648156

Subgroups — C2×#22550093 #30648156

Order 16256.

564 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 157 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14523233C211yesmaximal, characteristic
2C2×#16861198C211yesmaximal, characteristic, complemented
3C2×D4×Dic127C211yesmaximal, characteristic, complemented
4#22550093C218yesmaximal, direct product factor
5C2×Dic127⋊C8C211yesmaximal, characteristic, complemented
6C2×#16861133C211yesmaximal, characteristic, complemented
7C2×#16861131C211yesmaximal, characteristic, complemented
8C22×SD16×C127C211yesmaximal, characteristic, complemented
9C22×D508C2211yescharacteristic
10C23×Dic127—21no—
11C2×#14523233—24no—
12C2×#14523233C2214yes—
13C22×C127⋊C8C2211yescharacteristic
14#16861198—22no—
15#16861198C2214yescomplemented
16D4×Dic127—22no—
17D4×Dic127C2214yescomplemented
18C2×C4⋊Dic127C2211yescharacteristic
19C2×C22⋊Dic127—21no—
20C2×(C2×D127)⋊C4—21no—
21C2×(C2×D127)⋊C4—21no—
22C2×C4×Dic127C2211yescharacteristic
23Dic127⋊C8C2214yescomplemented
24#16861133C2214yescomplemented
25#16861131C2214yescomplemented
26C22×Q8×C127C2211yescharacteristic
27C22×D4×C127C2211yescharacteristic
28C2×SD16×C127—24no—
29C2×SD16×C127C2214yescomplemented
30C22×C8×C127C2211yescharacteristic
31C23×D127—21no—
32C2×D508—22no—
33C2×D508—21no—
34C2×D508—22no—
35C2×D508C2312yes—
36C22×Dic127—24no—
37C22×Dic127—21no—
38C22×Dic127—44no—
39C22×Dic127D411yescharacteristic
40C22×Dic127D411yescharacteristic
41#14523233—28no—
42C2×C127⋊C8—22no—
43C2×C127⋊C8C2312yes—
44C4⋊Dic127—21no—
45C4⋊Dic127C2312yes—
46C22⋊Dic127—24no—
47(C2×D127)⋊C4—24no—
48(C2×D127)⋊C4—24no—
49C4×Dic127D412yes—
50C4×Dic127C2312yes—
51C24×C127—21no—
52C2×Q8×C127—21no—
53C2×Q8×C127—22no—
54C2×Q8×C127—22no—
55C2×Q8×C127C2312yes—
56C2×D4×C127—21no—
57C2×D4×C127—22no—
58C2×D4×C127D414yes—
59C2×D4×C127C2312yes—
60C22×C4×C127—21no—
61C22×C4×C127C2311yescharacteristic
62SD16×C127—28no—
63C2×C8×C127—22no—
64C2×C8×C127C2312yes—
65C22×D127—22no—
66C22×D127—42no—
67C22×D127—42no—
68C22×D127—42no—
69D508—22no—
70D508—22no—
71D508—42no—
72C2×Dic127—22no—
73C2×Dic127—22no—
74C2×Dic127—22no—
75C2×Dic127—48no—
76C2×Dic127—42no—
77C2×Dic127C2×D412yes—
78C2×Dic127C2×D412yes—
79C127⋊C8—22no—
80C23×C127—24no—
81C23×C127—22no—
82C23×C127—42no—
83C23×C127—42no—
84C23×C127C2×D411yescharacteristic
85Q8×C127—22no—
86Q8×C127—22no—
87Q8×C127—42no—
88D4×C127—24no—
89D4×C127—22no—
90D4×C127C2×D414yes—
91C2×C4×C127—22no—
92C2×C4×C127—42no—
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2×D412yes—
95C2×C4×C127#4112yes—
96C2×C4×C127C2411yescharacteristic
97C8×C127—22no—
98C22×C127—24no—
99C22×C127—42no—
100C22×C127—42no—
101C22×C127—48no—
102C22×C127#12112yes—
103C22×C127#13512yes—
104C22×C127AGL(1,ℤ/8)12yes—
105C22×C127C22×D411yescharacteristic
106C2×D127—42no—
107C2×D127—42no—
108C2×D127—42no—
109C2×D127—84no—
110C4×C127—42no—
111C4×C127#12112yes—
112C4×C127C22×D411yescharacteristic
113C4×C127C2×#4111yescharacteristic
114Dic127—22no—
115Dic127—22no—
116Dic127—42no—
117C2×C127—44no—
118C2×C127#46314yescomplemented
119C2×C127C2×#12111yescharacteristic
120C2×C127C2×#13511yescharacteristic
121C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
122D127—82no—
123C2×#463—1271nomaximal, Sylow
124C127C2×#46311yescharacteristic, complemented, Sylow
125C22×SD16D12711yescharacteristic, complemented
126C22×D8—1271no—
127C2×#124—1271no—
128C2×C4×D4—1271no—
129#463—1278no—
130C2×C4⋊C8—1271no—
131C2×#103—1271no—
132C2×#102—1271no—
133C22×Q8C2×D12711yescharacteristic
134C22×D4—1271no—
135C22×D4C2×D12711yescharacteristic
136C23×C4—2541no—
137C2×SD16—24no—
138C2×SD16C2×D12714yescomplemented
139C2×D8—1274no—
140C2×D8—2544no—
141C22×C8—1271no—
142C22×C8C2×D12711yescharacteristic
143#124—1274no—
144#124—2542no—
145C4×D4—1274no—
146C4×D4—2542no—
147C2×C4⋊C4—1271no—
148C2×C22⋊C4—2541no—
149C2×C22⋊C4—2541no—
150C2×C22⋊C4—2541no—
151C2×C42—1271no—
152C4⋊C8—1274no—
153#103—1274no—
154#102—1274no—
155C24—21no—
156C24—2541no—
157C2×Q8—21no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8C22×D12712yes—
161C2×D4—21no—
162C2×D4—22no—
163C2×D4—1272no—
164C2×D4—2541no—
165C2×D4—2542no—
166C2×D4—2542no—
167C2×D4C22⋊D12714yes—
168C2×D4C22×D12712yes—
169C22×C4—21no—
170C22×C4—1271no—
171C22×C4—1271no—
172C22×C4—5084no—
173C22×C4—2544no—
174C22×C4—2541no—
175C22×C4C22×D12711yescharacteristic
176SD16—28no—
177D8—2548no—
178C2×C8—22no—
179C2×C8—1272no—
180C2×C8—2542no—
181C2×C8C22×D12712yes—
182C4⋊C4—1272no—
183C4⋊C4—2541no—
184C22⋊C4—2544no—
185C22⋊C4—2544no—
186C22⋊C4—2544no—
187C42—1272no—
188C42—1272no—
189C23—22no—
190C23—24no—
191C23—42no—
192C23—42no—
193C23—5082no—
194C23—5082no—
195C23—5082no—
196C23—2542no—
197C23D4×D12711yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—42no—
201D4—22no—
202D4—24no—
203D4—5082no—
204D4—2542no—
205D4—2542no—
206D4C2×C22⋊D12714yes—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—1272no—
210C2×C4—1272no—
211C2×C4—5082no—
212C2×C4—5088no—
213C2×C4—2542no—
214C2×C4—2542no—
215C2×C4—2542no—
216C2×C4D4×D12711yescharacteristic
217C2×C4C2×C22⋊D12712yes—
218C2×C4#1452324712yes—
219C2×C4C23×D12711yescharacteristic
220C8—22no—
221C8—2542no—
222C22—24no—
223C22—48no—
224C22—42no—
225C22—42no—
226C22—10164no—
227C22—5082no—
228C22—5082no—
229C22—5082no—
230C22#1686118112yes—
231C22#1686123012yes—
232C22#1686123612yes—
233C22C2×D4×D12711yescharacteristic
234C4—42no—
235C4—5082no—
236C4—2542no—
237C4—2542no—
238C4#1686118112yes—
239C4C22×C22⋊D12711yescharacteristic
240C4C2×#1452324711yescharacteristic
241C2—44no—
242C2—10162no—
243C2#2255009314yesdirect product factor
244C2C2×#1686118111yescharacteristic
245C2C2×#1686123011yescharacteristic
246C2C2×#1686123611yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×SD16×C127C2
socleC23×C127C2×D4
prime radicalC22×#14523233C2
hypercenterC22×SD16D127
hypocenterC127C2×#463
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#16861198C2semidirect
3. C2×D4×Dic127C2semidirect
4. #22550093C2direct (N × H)
5. C2×Dic127⋊C8C2semidirect
6. C2×#16861133C2semidirect
7. C2×#16861131C2semidirect
8. C22×SD16×C127C2semidirect
15. #16861198C22semidirect
17. D4×Dic127C22semidirect
23. Dic127⋊C8C22semidirect
24. #16861133C22semidirect
25. #16861131C22semidirect
29. C2×SD16×C127C22semidirect
118. C2×C127#463semidirect
124. C127C2×#463semidirect
125. C22×SD16D127semidirect
138. C2×SD16C2×D127semidirect
243. C2#22550093direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#145232332
C2×#168611982
C2×D4×Dic1272
#225500932
C2×Dic127⋊C82
C2×#168611332
C2×#168611312
C22×SD16×C1272
C2×#463127